Pentagonal geometries (Joint work with Terry S. Griggs and Tony Forbes, The Open University, UK.)

Generalized polygons are Bruhat-Tits buildings of rank two. They can also be defined in terms of their bipartite incidence graph, which has the property that the girth is twice the diameter. By the Feit-Higman Theorem (1964), the only finite generalized polygons are thin (two points on each line or two lines on each point) or the diameter is 3, 4, 6 or 8, corresponding to the finite projective planes, the generalized quadrangles, the generalized hexagons and the generalized octagons, respectively. In particular there are no generalized pentagons. An alternative way to generalize the pentagon was introduced by Simeon Ball et al. in [1]. In this talk I will discuss what we know about these incidence geometries. [1] S. Ball, J. Bamberg, A. Devillers and K. Stokes. An alternative way to generalize the pentagon. J. Combin. Des., 21:163â 179, 2013. [2] T. S. Griggs and K. Stokes. On pentagonal geometries with block size 3, 4 or 5. Springer Proc. in Math. & Stat., 159:147â 157, 2016.Non UBCUnreviewedAuthor affiliation: University of SkovdeOthe